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This thread is designed to be sort of an offshoot of LaLush's thread on macro (http://www.teamliquid.net/forum/viewmessage.php?topic_id=191702). In this thread, LaLush spends most of his time comparing SC2 to SC:BW. I am not going to spend any time talking about Brood War at all. Instead, I want to clear up the misconception of "Maximum Saturation". The current popular belief is that a base is optimally saturated when there are 3 workers for every mineral patch. Since there are 8 mineral patches at a standard base, this puts the "optimal" number of workers at 24. This is incorrect.
Craftonomics 101 – The Nexus is a Firm!
Note: I play protoss, so all my analogies will be protoss-related. However, everything I say applies to all three races, not just protoss.
Starcraft 2 is an economy-based strategy game. As such, basic economic principles can be applied. Let's say that my nexus is a beer factory. It requires money (minerals) to produce beer, and it receives income by selling beer (in Starcraft, income is received by mining minerals, so in this case selling beer is an analogy to mining minerals.) In Starcraft, the most conventional way to measure revenue is as a function of time. When watching a reply, the "Income" tab shows you the amount of minerals being received every minute (game time, not real time). Because workers mine at a certain rate of time, it makes the most sense to measure income as a function of time.
In economics, we say that the marginal revenue received by a firm is the amount of money it gets from producing one additional product. In Starcraft, the marginal revenue received by the player is the additional minerals they get from producing one additional worker. Conversely, the marginal cost incurred by a firm is the cost incurred by producing that one extra worker.
Now here's the most important part: The firm maximizes its revenue when the marginal revenue received from making one extra worker is equal to the marginal cost of producing that worker.
Sounds simple enough, right?
Craftonomics 201 – The Marginal Cost is Zero.
That's right, it is impossible to find the marginal cost of producing one extra worker. Now I'm sure most of you are saying "No it's not! Workers cost 50 minerals, so the marginal cost of making one extra worker is 50 minerals!"
This is an easy mistake to make for one very simple reason: revenue is measured as a function of time. So if we are going to try to equate marginal revenue and marginal cost, marginal cost must also be a function of time.
Okay, so let's try this. How do we change a cost of 50 minerals into a function of time? Well that should be simple enough, right? A worker takes 17 seconds to build, and 17 seconds is 0.283 minutes, and so we get:
50 minerals / 0.283 minutes = 176.68 minerals per minute
But hold on a second there. This equation represents the per minute cost of producing a worker – in other words, if I make one extra probe, it's going to cost me 176.68 minerals per minute, meaning that if I make that probe, it's going to lower my income by 176.68 minerals per minute. This is obviously not right. In order to get the correct marginal cost per minute, we need to know how long the workers will be in existence for.
The reason for this is simple: Producing X amount of workers costs 50*X minerals, where 50*X is the total cost incurred by the player from producing workers. In order to derive the total cost per minute, we need to know how long the game lasts in minutes. So the total cost function, just like the revenue function, is a function of time:
Total cost = 50X / T, where T = time in minutes
To get the marginal cost, we simply take the derivative of the total cost function with respect to X:
(50X / T)' = 50/T
Now here's the kicker: since we don't know how long the game will last, we should investigate what happens to the marginal cost function as the game gets longer and longer. Let's see what happens when the game lasts 5 minutes, when the game lasts 10 minutes, and when the game lasts 20 minutes:
Marginal Cost when T = 5: 50/5 = 10 minerals per minute
Marginal Cost when T = 10: 50/10 = 5 minerals per minute
Marginal Cost when T = 20: 50/20 = 2.5 minerals per minute
We're starting to see a trend here: as the game length increases, the marginal cost of producing another worker decreases. Since we cannot predict how long a game will last, we must define the marginal cost in terms of the limit: the behavior of a function (in this case, the marginal cost function) as it approaches a certain value. Since we do not know how long a game will last, we want to define the marginal cost function as the game time gets infinitely greater:
lim(50/T) as T → ∞ = 0
This is saying that the limit of the marginal cost function, as time gets infinitely greater, is zero. Thus, the expected marginal cost of producing an additional probe is zero.
Craftonomics 101 Revisited: Setting Marginal Revnue equal to Marginal Cost
Now that we have established that the marginal cost is zero, setting marginal revenue equal to marginal cost should be very easy. First, we need to derive the marginal revenue. This is very easy, and many people have done this before. However, Starcraft's income measuring system is very inaccurate – it only measures the income received per minute in increments of 20. The true change in revenue received by producing one extra worker is unlikely to be rounded off so nicely. So, to get an estimate of the true marginal revenue, I generated predicted values of income per minute as a cubic function of the number of workers currently mining. Don't get too hung up on the method, it's actually very simple, and should be a relatively accurate prediction of the true marginal revenue function. The following is a chart of the values I got:
![[image loading]](http://i53.tinypic.com/bi6pm9.jpg)
To help interpret this, take a look at quantity 14. The chart says that having 14 workers mining minerals nets you 572.96 minerals per minute, and having 14 workers instead of 13 gets you an additional 35.59 minerals per minute. Here is a graphical representation (keep in mind, I didn't calculate values before worker number 7, so everything before that is set to zero):
![[image loading]](http://i52.tinypic.com/xpomtj.png)
Now that we have all this, we can set the marginal revenue equal to marginal cost – this is where our economy is maximized. Since our marginal cost is zero, we want to produce workers until the marginal revenue received from producing an extra worker is zero. As we can see in the chart above, the marginal revenue is positive until we make the 28th worker. Once we have 27 workers, if we make the 28th, our marginal revenue will turn negative, and our total income per minute will go down. And so we can see that the optimal saturation level is 27 workers, not 24.
Lessons from Craftonomics
Our previous perception that 3 workers per mineral patch equates to optimal saturation is wrong. We should define optimal saturation in terms of producing workers until there are no longer minerals to be gained by doing so, not in terms of workers per mineral patch. When we apply economic concepts to deriving optimal saturation, we find that 27 workers constitutes optimal saturation.
An edit in response to some posts: This is purely about how to get the most minerals out of one base. Please do not think that I am advocating staying on one base over expanding, because I'm not. Please stop saying that I am wrong because expanding is better than staying on one base - this is not what I'm talking about here.
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It really sucks to have to say this... but all your extra work and fancy graphs come to an incorrect conclusion. You can pretty up the argument all you want, adding all sorts of neat presentation styles, but 3 * 8 = 24. And if additionalIncome(x) 24 <- x = 0, then your 4 additional workers over 24 do nothing but take up pop space and require 200 minerals. I would guess that you have either a rounding or a logic error in your thorough.. but incorrect analysis.
Or you can forget the deductive approach and just do it empirically.
Mine a base till you have 24 workers. Make a new base.
Have 24/28 workers mine that base for your two trials.
After 2 minutes of your workers "settling", see how many minerals 24 workers v 28 mine over 5 minutes.
You will realize that there is no statistically significant difference over 40 trials... as has been shown on numerous occasions.
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What's your justification for using a cubic function re: revenue estimation by worker?
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Btw, the cost is actually wrong it's 50 minerals for the worker +100/8 = 12.5 for the supply. The real cost is 62.5 pr worker.
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Well, the problem with this analysis is that it assumes that the value of money or minerals is static, which isn't the case. In real world economics, you have to consider opportunity cost, i.e. when you invest X amount of money into Y, you lose out on the profit that could be made by investing X into something else.
In "craftonomics" this also holds true, as the money spent on a probe could also be spent on something else and maybe that something else would increase your chances of winning more than getting the extra revenue of one drone.
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So, to get an estimate of the true marginal revenue, I generated predicted values of income per minute as a cubic function of the number of workers currently mining. Don't get too hung up on the method, it's actually very simple, and should be a relatively accurate prediction of the true marginal revenue function. So, the more workers I have (above 27), the lower my income?
So if my nat is harassed by mutas and I evacuate the probes to my main, I will get negative income due to oversaturation? Doesn't make much sense to me.
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I think the issue here is that, the op justified that making 27 workers still yields an increase in income/gains minerals from doing so, but the gain in making 3 extra workers is marginal and not worth the 150 minerals invested in doing so. The marginal revenue of the last 3 workers is 8.55, 5.55 and 2.46, better to have them in the nat
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Emperor_Earth is right. It's too bad that you put in a lot of time and effort (plus outside knowledge from microeconomics) and 8*3 is still 24. I really think the extra 150mins is going to be a waste if we're talking about cost-effectiveness and saturation.
I just don't understand what those 3 workers are doing (aside from finding an open mineral patch) when all the patches are saturated
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I am wondering if you have the right definition for "Optimal Saturation". Here, you define optimal saturation as the maximum rate of mineral collection.
However, what if the definition should answer the question: What is the number of workers that I need to get in order to support the build order? This is important if one wants to hit critical timing. Note that under this definition, one could be over-saturated under this definition below 3 workers per patch. To see this, think about cutting workers in the optimal 4-gate build for Protoss.
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You will realize that there is no statistically significant difference over 40 trials... as has been shown on numerous occasions.
Statistical significance has nothing to do with anything here. If I have one base mining with 28 workers, and one base mining with 24, and the base with 28 workers gets me an extra 20 minerals per minute, I'll take the base mining with 28 minerals. Do you care if 20 minerals is statistically significant? I know I don't. This is a common misapplication of statistical significance; it shouldn't apply here. When we are talking about minerals, we are talking about absolutes.
What's your justification for using a cubic function re: revenue estimation by worker?
Diminishing marginal returns. It's not a linear function, and the difference between a cubic and a quadratic function to fit the data is almost non-existent.
Btw, the cost is actually wrong it's 50 minerals for the worker +100/8 = 12.5 for the supply. The real cost is 62.5 pr worker.
Please explain, I don't understand how you come to this conclusion. I don't even know what your unit of measurement is.
In "craftonomics" this also holds true, as the money spent on a probe could also be spent on something else and maybe that something else would increase your chances of winning more than getting the extra revenue of one drone.
I agree completely. Notice that I never advocate rushing to get 27 workers as soon as possible. I only say that 27 is where you get the highest income per minute. That's all. If at some point your macro slips, and you end up having 400 minerals lying around (as often happens to people) and you have 24 probes on minerals, invest in a few more to get that optimal 27. This would be an example of a very low opportunity cost. But obviously, if you are about to face a large push and need units now, don't pump more probes. That would be a good example of a high opportunity cost.
So if my nat is harassed by mutas and I evacuate the probes to my main, I will get negative income due to oversaturation? Doesn't make much sense to me.
This isn't at all what I'm saying. I'm saying that if you have more than 27 probes on a single base, your INCOME PER MINUTE will start to decrease. So instead of getting, say, 800 minerals per minute with 27 probes, you will be getting maybe 795 per minute with 28. You are still getting positive income, but the rate at which you get it starts to decrease.
Good questions so far guys, keep them coming.
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All this ignores the game itself. You always want to make workers past 24 simply because you expect (I hope) to expand soon and just like building an extractor and immediately dedicating 2 or 3 workers to it, there is no point expanding without gaining immediate extra production (aka - have excess workers ready and if necessary just wasting their time).
Also under pressure the first thing to slip might be worker production (yours or theirs) so making excess will help you maintain saturation even when getting harassed.
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Emperor_Earth is right. It's too bad that you put in a lot of time and effort (plus outside knowledge from microeconomics) and 8*3 is still 24. I really think the extra 150mins is going to be a waste if we're talking about cost-effectiveness and saturation.
Regarding "all this time I spent" - this took me about 20 minutes to derive the numbers, and about 20 minutes to write up. It was a nice break from the vast amount of activities that take up most of my day.
Secondly, I have no idea what "8x3=24" is supposed to mean. Is this some universal fact of starcraft? You are spitting out things that other people told you, and completely discounting everything I said simply because it is different from "8x3=24". I have just provided you with empirical evidence that 27 workers gets you more income per minute than does 24. Do you have evidence that says otherwise? Please show it to me.
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I am wondering if you have the right definition for "Optimal Saturation". Here, you define optimal saturation as the maximum rate of mineral collection.
You can't change my own definition and then tell me what I said was wrong, lol. That's like your professor telling you that 0.99999999999 is equal to one, and him showing you a mathematical proof, but then you raise your hand and say "I wonder if your definition of one is correct. If one is actually two, then 0.999999999 does not equal two."
You got it right, optimal saturation by my definition is the most minerals that can be collected per minute. I said nothing about build orders.
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If the model was more complete, it would show expanding before 27 workers is far better than 27 workers on one base. If you are confined to one base and know this will be true for some time, it is worth it to go to 27 workers. But that could be shown just by saying the maximum income is for that number, no need to invoke economics or fancy graphs.
Both 27 and 24 are wrong. You should produce workers continuously in expectation of expanding as early as possible, up to about 70 workers. I don't see many situations where that doesn't hold.
Oh, and optimal's more usual meaning is most likely to win games. Expanding five times before gateway will produce maximum income but is, I'm sure you will agree, not optimal in any Starcraft II game.
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In Starcraft, every worker number >3 on a mineral patch does not increase your mineral count since they only walk around from one patch to another. Really just a fact. Therefor 24 is the optimum total saturation on minerals.
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If the model was more complete, it would show expanding before 27 workers is far better than 27 workers on one base. If you are confined to one base and know this will be true for some time, it is worth it to go to 27 workers. But that could be shown just by saying the maximum income is for that number, no need to invoke economics or fancy graphs.
Both 27 and 24 are wrong. You should produce workers continuously in expectation of expanding as early as possible, up to about 70 workers. I don't see many situations where that doesn't hold.
Oh, and optimal's more usual meaning is most likely to win games. Expanding five times before gateway will produce maximum income but is, I'm sure you will agree, not optimal in any Starcraft II game.
I said right at the beginning of the post that my definition of optimal saturation was the amount of workers that gets you the most income per minute on one base. That's it. Nothing more. This isn't some huge guide on how to win games more often.
Also, if you could please direct me to where I said that staying on one base with 27 workers is better than expanding, that would be great, because I don't recall saying anything about expanding as opposed to staying on one base.
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In Starcraft, every worker number >3 on a mineral patch does not increase your mineral count since they only walk around from one patch to another. Really just a fact. Therefor 24 is the optimum total saturation on minerals.
It's just fact? Where are you getting this "fact" from? Because I ran a simulation comparing 24 workers to 27, and 27 was netting me more minerals. Can you offer any evidence to back up your fact?
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United States5162 Posts
Wow so many people get hung up on the fact that its commonly accepted that three workers per patch is optimal saturation.
The above is true. However, as this work shows, and really as it's shown in the Analysis of Macro thread as well, 24 works on minerals in not the maximum saturation. Putting more workers up to about 30 results in a gain of income per minute. The reason it's generally accepted as 24 workers is saturation is because the difference between 24 workers and 30 workers is quite small.
What this has shown beyond the Analysis of Macro thread is that while 30 workers does produce more income than 27 by a small amount, the economic cost of those last 3 workers is not worth the investment while increasing workers from 24 to 27 is.
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The only thing I can say is... you put in work for a useless statistic for the game of SC2. The one thing that is ignored is expansions. Which then puts the optimal mining workers at 16. simply because:
Base 1 = 16 works
Base 2 = 11 works
Mines more minerals than:
Base 1 = 27 workers
Simply due to how the first 2 workers on a mineral patch mine FAR more minerals than any workers after that. Even taking into consideration the cost of a 2nd base, the mineral gains are still much higher my making a 2nd base.
Even if you look at your own graph... the margianl minerals per minute of the first 16 works > cost of a CC/Nexus/Hatch.
Basically, expanding at 16 on minerals > going to 27 workers... ALWAYS.
This basically just means that your work its sort of useless unless in the odd scenario where there is only 1 mining base left on map and you have X amount of workers, killing all but 27 will result in the highest mineral gains...
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Uh no offense, but marginal cost is the derivative of total cost with respect to quantity. Not time. There's a big difference between the two. I understand that you think MC lim 0 because you don't know how long the game will last. However that question comes more into play in the MR curve, not the MC curve. The longer the game lasts, the greater the MR, not the less the MC.
You're certainly correct that at economic optimization you want MC=MR, it's just that you're calculating MC incorrectly.
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On February 11 2011 07:22 kNightLite wrote:
Uh no offense, but marginal cost is the derivative of total cost with respect to quantity. Not time. There's a big difference between the two. I understand that you think MC lim 0 because you don't know how long the game will last. However that question comes more into play in the MR curve, not the MC curve. The longer the game lasts, the greater the MR, not the less the MC.
You're certainly correct that at economic optimization you want MC=MR, it's just that you're calculating MC incorrectly.
You're absolutely right in that this is a very unorthodox way of calculating MC. The problem is that you can't calculate it normally because this is a video game. So, one must make some assumptions and change things around a bit.
Perhaps a better way of defining the marginal cost here is to think of it as the marginal cost per minute. The reason is because MR is defined by the game itself as marginal revenue per minute. We need to put MC in the same unit of measurement. So if you calculate MC normally, and then set it as a function of time, and take the limit (as I did), you will see that the marginal cost per minute (which is what we are really measuring here) is in fact zero.
Good question.
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The only thing I can say is... you put in work for a useless statistic for the game of SC2. The one thing that is ignored is expansions. Which then puts the optimal mining workers at 16. simply because:
Base 1 = 16 works
Base 2 = 11 works
Mines more minerals than:
Base 1 = 27 workers
Simply due to how the first 2 workers on a mineral patch mine FAR more minerals than any workers after that. Even taking into consideration the cost of a 2nd base, the mineral gains are still much higher my making a 2nd base.
Even if you look at your own graph... the margianl minerals per minute of the first 16 works > cost of a CC/Nexus/Hatch.
Basically, expanding at 16 on minerals > going to 27 workers... ALWAYS.
This basically just means that your work its sort of useless unless in the odd scenario where there is only 1 mining base left on map and you have X amount of workers, killing all but 27 will result in the highest mineral gains...
Can you please point out where I said that rushing to 27 workers was better than expanding? Because I don't believe I ever said that.
Also, you are quick to point out how what I did was useless, but I was wondering if you could please tell me how you have contributed in any way to the community? You seem to be on an awfully high pedestal.
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While its an interesting read, there are several few ideas that are not really considered when calculating your model, and because i'm tired i'll just keep it brief. Simply put, you don't factor in opportunity cost very heavily here - which is the prime consideration when looking at the ideal number of workers to make - sacrificing attacking units / tech / additional expansions (to increase the MP of each additional workers, thus increasing efficiency) / what have you ends up being a large consideration - Maybe some time I'll write on this, but the eventual conclusion is that its highly situational and sometimes its simply better to expand to take advantage of the increase in MP of each worker than make 3 more workers for a small increase in Total Product, given that you have the ability to defend the expansion. Again, it gets complicated, but you need to consider the opportunity cost, perhaps measured in terms of "risk" or "perceived risk", which determines what applicable strategy is best sought after.
Oh economics, i love you.
Or, TLDR - its pretty fucking difficult coming up with an applicable economic model making conclusions without ignoring important assumptions in starcraft, so all conclusions found need some sort of asterisk.
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Let me just take Drunken.Jedi's comment regarding tradeoff and elaborate a little on the conclusions of the op.
I have to say that even if the numbers are correct, I do not believe your analysis is enough to prove that "27 workers constitutes optimal saturation." I actually believe you just proved otherwise, that 24 workers constitutes optimal saturation.
Since we are being analytical here, let's step back for a moment and get Webster's definition for optimal: "the best or most favorable point; most favorable or desirable".
When you take the concept of mineral patch saturation, the discussion is obviously taking place in the context of Starcraft II and taking into account real and practical applications, otherwise it is just mindless theorycrafting. Although this may seem like nitpicking, I actually believe it is not, as the usefulness of Stacraft II related definitions directly helps subside every other single discussion, may it be here in the TL forums or not, regarding the game.
That being said, I would kindly ask for any example whatsoever of a real game situation in which it is optimal for any player to go beyond 24 workers to get the supposedly optimal 27. Your example was "If at some point your macro slips [...]" - well, that is just not optimal, is it? I am most certain that in the tradeoff between building the 25th worker in your base and building another unit or building, it will always be truly optimal to do the latter.
All that just to say that your analysis proves that the gain from any worker beyond the 24th is so small that it should be optimal to just do something else with those 50 minerals and 1 food. Although mathematically the conclusion may be true, it is just useless in the context of the discussion.
In my opinion, 24 workers still constitutes optimal saturation.
Edit: let me take the "useless" back, and change it to counterproductive. I do not mean to just critic the op's very cool work, just point out my opinion on the need have productive definitions to discuss the game.
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Again, it gets complicated, but you need to consider the opportunity cost, perhaps measured in terms of "risk" or "perceived risk", which determines what applicable strategy is best sought after.
I addressed opportunity cost in a previous post. The problem with what you are proposing is how exactly do you measure risk? You, along with several other people, seem to think that I am advocating spending all resources on workers until you get 27 workers. This is simply not the case, and I never said this anywhere. All I said was that 27 workers gets you more income than 24. I didn't say HOW to get 27 workers, I didn't say WHEN to get 27 workers, I just said that 27 workers gets you a higher income per minute than does 24.
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Thread resolved: http://www.teamliquid.net/forum/viewmessage.php?topic_id=192072¤tpage=5#96
You need to stop being so hostile. It's obvious to me that you're trying to put yourself in a teacher role where you believe you contributed in some meaningful way and that the rest of us just have it wrong. I understand you just completed or are in the process of completing your Introduction to Economics class, but you need to understand that your approach, while notable, is fundamentally flawed. Why? Because your conclusion is based upon the formula you used to determine the marginal revenue.
Why does this produce false information?
Because I know whatever formula you used does not take into account SCV travel time between mineral patches and the CC, and between each of the mineral patches when said patches are occupied.
The fundamental flaw here is that your model does not take into account the factors which require a much more complicated mathematical skill set than what I believe you have based on your post. (If you're a PhD in mathematics, please correct me)
Realize the mechanics of the game: when occupied by two workers timed correctly, a mineral patch only has a few milliseconds, if not zero ms, when they are not occupied and thus a third worker would occupy the patch. Three workers per patch is already pushing it, and any more is just overkill. Now I'm not saying you have to match the results of your theory to what would be expected, but you have to realize that your work is not complete and you cannot come stomping into the forums demanding you be heard if your calculations do not take into account ALL of the factors at play during mining time.
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All that just to say that your analysis proves that the gain from any worker beyond the 24th is so small that it should be optimal to just do something else with those 50 minerals and 1 food. Although mathematically the conclusion may be true, it is just useless in the context of the discussion.
I don't mean to sound rude here, but I honestly don't see what is so difficult to grasp about this post. You keep trying to bring in "contexts" and "situations" when all I did was show that 27 workers gets you more income than 24 does. If you have the option of getting 27 workers instead of just 24, then you should take it. If you don't have that option (because of the situation or the context or whatever), then stay at 24. You said it yourself: mathematically speaking, 27 gets you more than 24. That's all I showed here.
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On February 11 2011 07:29 natewOw wrote:Show nested quote +The only thing I can say is... you put in work for a useless statistic for the game of SC2. The one thing that is ignored is expansions. Which then puts the optimal mining workers at 16. simply because:
Base 1 = 16 works
Base 2 = 11 works
Mines more minerals than:
Base 1 = 27 workers
Simply due to how the first 2 workers on a mineral patch mine FAR more minerals than any workers after that. Even taking into consideration the cost of a 2nd base, the mineral gains are still much higher my making a 2nd base.
Even if you look at your own graph... the margianl minerals per minute of the first 16 works > cost of a CC/Nexus/Hatch.
Basically, expanding at 16 on minerals > going to 27 workers... ALWAYS.
This basically just means that your work its sort of useless unless in the odd scenario where there is only 1 mining base left on map and you have X amount of workers, killing all but 27 will result in the highest mineral gains... Can you please point out where I said that rushing to 27 workers was better than expanding? Because I don't believe I ever said that. Also, you are quick to point out how what I did was useless, but I was wondering if you could please tell me how you have contributed in any way to the community? You seem to be on an awfully high pedestal.
Can you point out a single time in game where this will EVER be useful? Can you point out a time in the game that you will ever concievebly want 27 works on any one base when you have another base open for expansion?
It is all fine and dandy that 27 workers = where the marginal cost = 0... but its not something you will ever use in game.
You WILL on the other hand expand and keep 16 workers per base simply because the MC is highest for those first 16 workers at any base.
In a one basing situtation you will either:
a) Cut workers for your build = does not hit 27 workers
b) Continue to make workers upto 44 workers at the high end so they can be maynarded over to an expansion.
In game, you will NEVER go "man... I really need to get some more income, time to get up to 27 workers on 1 base" you will on the other hand go "man... I really need some more income... time to expand" simply because the mineral output of an expansion is greater then gettting any where near 27 workers on a single base.
- - -
This is interesting yes, but it is useless for the game of SC2 as it will never be used.
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Even if you're theory is correct, there are several large practical concerns which will likely prevent this from being useful. Several people have already mentioned expansion timing. However, the biggest concern that I have is whether the additional income you gain from workers 25-27 overcomes the cost before mineral patches start becoming mined out.
The reason you didn't come across this in your analysis is that you asked how long the worker will be in existence for, not how long the worker will be USEFUL for. I don't know the timing exactly, but it seems quite likely that if the marginal revenue from the 27th worker is LESS THAN 10 MINS PER MINUTE (as indicated in your graph) that the mineral patches would be long mined out before that worker mined for the 5-10 minutes necessary to regain its cost.
EDIT: the table gives the 27th worker as providing an additional income of 2.46 minerals per minute, so it would need to mine for a whopping 20 minutes to regain the 50 minerals spent on it
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On February 11 2011 07:41 natewOw wrote:Show nested quote +All that just to say that your analysis proves that the gain from any worker beyond the 24th is so small that it should be optimal to just do something else with those 50 minerals and 1 food. Although mathematically the conclusion may be true, it is just useless in the context of the discussion. I don't mean to sound rude here, but I honestly don't see what is so difficult to grasp about this post. You keep trying to bring in "contexts" and "situations" when all I did was show that 27 workers gets you more income than 24 does. If you have the option of getting 27 workers instead of just 24, then you should take it. If you don't have that option (because of the situation or the context or whatever), then stay at 24. You said it yourself: mathematically speaking, 27 gets you more than 24. That's all I showed here.
Fair enough, I'll simply ask this - while MR is positive, shouldn't it be considered that the cost of adding such small minutely revenues is not particularly logical barring some sort of inability to spend money, or in other words, the time it takes for the Total revenue for that individual worker may not be worthwhile for the total marginal cost of that worker, especially when considering the time value of money (aka money is worth more than later)? Or otherwise, perhaps it's never logical to get 27 workers in a mineral patch.+ Show Spoiler + (which I don't believe, you just have to create reasoning for it, such as the potential for the loss of mining, the usage of defence, expanding to other expansions).
However, this is kind of meaningless, I'm just talking for the sake of talking. I don't really disagree with your conclusions.
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You need to stop being so hostile. It's obvious to me that you're trying to put yourself in a teacher role where you believe you contributed in some meaningful way and that the rest of us just have it wrong. I understand you just completed or are in the process of completing your Introduction to Economics class, but you need to understand that your approach, while notable, is fundamentally flawed. Why? Because your conclusion is based upon the formula you used to determine the marginal revenue.
I'm being hostile? I was under the impression that people were attacking my work (thus far without merit, I have yet to see a single post that has undermined anything I said) and I was defending it. It's a standard academic procedure. You criticize, I defend.
Also, if I am wrong, can you please provide evidence that 24 workers gets you more minerals per minute than does 27?
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On February 11 2011 07:45 natewOw wrote:Show nested quote +You need to stop being so hostile. It's obvious to me that you're trying to put yourself in a teacher role where you believe you contributed in some meaningful way and that the rest of us just have it wrong. I understand you just completed or are in the process of completing your Introduction to Economics class, but you need to understand that your approach, while notable, is fundamentally flawed. Why? Because your conclusion is based upon the formula you used to determine the marginal revenue. I'm being hostile? I was under the impression that people were attacking my work (thus far without merit, I have yet to see a single post that has undermined anything I said) and I was defending it. It's a standard academic procedure. You criticize, I defend. Also, if I am wrong, can you please provide evidence that 24 workers gets you more minerals per minute than does 27?
No one is attacking the fact that 27 workers mines more minerals than 24 workers... we are attacking the usefullness of this fact.
If you cannot explicitly tell us when you will ever purposefully get 27 workers on a sinlge base when there are availiable expansions still unmined, please by all means let me know and I will resend any doubt I had about your fact and will even add it to my signature so that all can see that we should get 27 workers on a single mining base in scenario X.
If you cannot give an scenario for its usefullness, then it is in fact a useless fact and adds nothing to the game and its just a fun little fact to know but impacts nothing gameplay wise.
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On February 11 2011 06:44 Drunken.Jedi wrote:
Well, the problem with this analysis is that it assumes that the value of money or minerals is static, which isn't the case. In real world economics, you have to consider opportunity cost, i.e. when you invest X amount of money into Y, you lose out on the profit that could be made by investing X into something else.
In "craftonomics" this also holds true, as the money spent on a probe could also be spent on something else and maybe that something else would increase your chances of winning more than getting the extra revenue of one drone.
Oh God finally someone else who sees SC2 choices in terms of opportunity cost. Once I tried discussing it with Cloud and he thought I was trolling him -_-"
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On February 11 2011 07:01 natewOw wrote:Show nested quote +Btw, the cost is actually wrong it's 50 minerals for the worker +100/8 = 12.5 for the supply. The real cost is 62.5 pr worker. Please explain, I don't understand how you come to this conclusion. I don't even know what your unit of measurement is.
You have to build a supply depot/overlord/pylon to be able to build a worker, that is where the extra cost of 12.5 minerals comes from.
I do wonder how you ended up with going towards infinity in an equation with a set amount of minerals on a particular map. The biggest maps have about 14 bases with around 8 patches each, at 1500 minerals for a total of 168.000 minerals total.
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The point everyone else is making is that the situation in which you would actually choose to get more than 24 workers per base to exactly 27 is just plain nonexistent, and thus rendering your little post here utterly null
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On February 11 2011 07:45 natewOw wrote:Show nested quote +You need to stop being so hostile. It's obvious to me that you're trying to put yourself in a teacher role where you believe you contributed in some meaningful way and that the rest of us just have it wrong. I understand you just completed or are in the process of completing your Introduction to Economics class, but you need to understand that your approach, while notable, is fundamentally flawed. Why? Because your conclusion is based upon the formula you used to determine the marginal revenue. I'm being hostile? I was under the impression that people were attacking my work (thus far without merit, I have yet to see a single post that has undermined anything I said) and I was defending it. It's a standard academic procedure. You criticize, I defend. Also, if I am wrong, can you please provide evidence that 24 workers gets you more minerals per minute than does 27?
What about the rest of my post? You clearly didn't want to acknowledge the rest of it because you wanted to take my first paragraph out of context.
Can you please provide me evidence that there isn't a flying spaghetti monster between Earth and Mars? You see what you did?
I'm saying that your work is incomplete because the formula you used to derive 27 workers is fundamentally flawed. Does it model worker movement, travel time between its destinations, the calculation the client uses to determine whether or not a worker moves or stays on a mineral patch if it is currently occupied? I didn't think so. I am not going to sit for a few days trying to determine it myself because I have better things to do, but I can assure you that no one has to do so in order to see that what you did is incomplete. Therefore your number of 27 is questionable AT BEST and PLAIN WRONG AT WORST.
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On February 11 2011 07:49 Insanious wrote:Show nested quote +On February 11 2011 07:45 natewOw wrote:You need to stop being so hostile. It's obvious to me that you're trying to put yourself in a teacher role where you believe you contributed in some meaningful way and that the rest of us just have it wrong. I understand you just completed or are in the process of completing your Introduction to Economics class, but you need to understand that your approach, while notable, is fundamentally flawed. Why? Because your conclusion is based upon the formula you used to determine the marginal revenue. I'm being hostile? I was under the impression that people were attacking my work (thus far without merit, I have yet to see a single post that has undermined anything I said) and I was defending it. It's a standard academic procedure. You criticize, I defend. Also, if I am wrong, can you please provide evidence that 24 workers gets you more minerals per minute than does 27? No one is attacking the fact that 27 workers mines more minerals than 24 workers... we are attacking the usefullness of this fact. If you cannot explicitly tell us when you will ever purposefully get 27 workers on a sinlge base when there are availiable expansions still unmined, please by all means let me know and I will resend any doubt I had about your fact and will even add it to my signature so that all can see that we should get 27 workers on a single mining base in scenario X. If you cannot give an scenario for its usefullness, then it is in fact a useless fact and adds nothing to the game and its just a fun little fact to know but impacts nothing gameplay wise.
all ins?
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On February 11 2011 07:27 natewOw wrote:Show nested quote +On February 11 2011 07:22 kNightLite wrote:
Uh no offense, but marginal cost is the derivative of total cost with respect to quantity. Not time. There's a big difference between the two. I understand that you think MC lim 0 because you don't know how long the game will last. However that question comes more into play in the MR curve, not the MC curve. The longer the game lasts, the greater the MR, not the less the MC.
You're certainly correct that at economic optimization you want MC=MR, it's just that you're calculating MC incorrectly. You're absolutely right in that this is a very unorthodox way of calculating MC. The problem is that you can't calculate it normally because this is a video game. So, one must make some assumptions and change things around a bit. Sorry, but no, it doesn't. In fact it's actually easier to build an economic model for SC2 than most real-world scenarios because of the relative absence of fixed costs, scaling, and externalities. Game length has an impact upon how many total minerals you receive over time. It doesn't change the fact that building an extra SCV is always going to cost 50 minerals. If you want to view saturation as a present value vs future value question that's fine, but that's totally separate from cost analysis.
Perhaps a better way of defining the marginal cost here is to think of it as the marginal cost per minute. The reason is because MR is defined by the game itself as marginal revenue per minute. We need to put MC in the same unit of measurement. So if you calculate MC normally, and then set it as a function of time, and take the limit (as I did), you will see that the marginal cost per minute (which is what we are really measuring here) is in fact zero.
Marginal cost per minute is a meaningless statistic. Especially when you let time approach infinity. Look at your own graph. At quantity = 1-6 your measure of MR = 0 just as much at 27.
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Fair enough, I'll simply ask this - while MR is positive, shouldn't it be considered that the cost of adding such small minutely revenues is not particularly logical barring some sort of inability to spend money, or in other words, the time it takes for the Total revenue for that individual worker may not be worthwhile for the total marginal cost of that worker, especially when considering the time value of money (aka money is worth more than later)? Or otherwise, perhaps it's never logical to get 27 workers in a mineral patch.
Sure. I mentioned that we want to investigate what happens to MC and MR as the game time, T, becomes greater. If we look at the marginal revenue provided by the 27th worker, the total revenue it provides is its total revenue provided less its total cost. I said that the total revenue provided by the 27th worker was about 2.46. So the total revenue provided by this worker is:
(2.46*T') - 50, where T' is the amount of time remaining in the game.
In order to find when it would be productive to get this 27th worker, we need to solve for T', which is the minimum amount of remaining game time in order for the 27th worker to start paying for itself. In this case, T' is about 20 minutes. After 20 minutes (game time, not real time), the probe will start netting you positive net revenue. Obviously this would only be useful in longer macro games.
So while the amount of marginal revenue provided by the 27th probe is very small, after 20 minutes or so of game time, it will start to pay for itself.
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On February 11 2011 07:51 Nagano wrote:Show nested quote +On February 11 2011 07:45 natewOw wrote:You need to stop being so hostile. It's obvious to me that you're trying to put yourself in a teacher role where you believe you contributed in some meaningful way and that the rest of us just have it wrong. I understand you just completed or are in the process of completing your Introduction to Economics class, but you need to understand that your approach, while notable, is fundamentally flawed. Why? Because your conclusion is based upon the formula you used to determine the marginal revenue. I'm being hostile? I was under the impression that people were attacking my work (thus far without merit, I have yet to see a single post that has undermined anything I said) and I was defending it. It's a standard academic procedure. You criticize, I defend. Also, if I am wrong, can you please provide evidence that 24 workers gets you more minerals per minute than does 27? What about the rest of my post? You clearly didn't want to acknowledge the rest of it because you wanted to take my first paragraph out of context. Can you please provide me evidence that there isn't a flying spaghetti monster between Earth and Mars? You see what you did? I'm saying that your work is incomplete because the formula you used to derive 27 workers is fundamentally flawed. Does it model worker movement, travel time between its destinations, the calculation the client uses to determine whether or not a worker moves or stays on a mineral patch if it is currently occupied? I didn't think so. I am not going to sit for a few days trying to determine it myself because I have better things to do, but I can assure you that no one has to do so in order to see that what you did is incomplete. Therefore your number of 27 is questionable AT BEST and PLAIN WRONG AT WORST.
Worker travel time and movement are simply components that go into determining the revenue per minute. I don't need to model these because blizzard did that already when they provided an income per minute meter. Why do I need to try and break it down when I have the final product?
Also, I'm still waiting for you to prove to me that 24 probes gets you more income per minute than 27. If you can't do that, please don't tell me I'm wrong.
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At quantity = 1-6 your measure of MR = 0 just as much at 27.
Read the post dude, I said that I didn't measure quantities 1-6, so I put in zero as placeholders.
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So now you're using the income per minute meter? I thought you acknowledged that it was inaccurate for the calculations you were doing?
Am I getting something wrong here? What did you use to calculate your numbers? The Blizzard meter or your own formula?
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I love how he is totally ignoring the fact that this little "27 > 24!!!!!!" isn't applicable in a real game
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The point everyone else is making is that the situation in which you would actually choose to get more than 24 workers per base to exactly 27 is just plain nonexistent, and thus rendering your little post here utterly null.
A huge amount of economic theory is based on assumptions that simply do not hold in the real world, and yet economics is one of the most studied sciences in the world. Just because it is not blatantly practical doesn't make it "utterly null".
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Maybe I'm missing something here.
My question is:
Why the heck is there a diminishing return before the 8th worker here? Let me know if I'm stupid as I have no experience in economics.
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On February 11 2011 07:56 natewOw wrote:
Also, I'm still waiting for you to prove to me that 24 probes gets you more income per minute than 27. If you can't do that, please don't tell me I'm wrong.
Not to mirror your hostility here, but can you please tell me where I said 24, or any other quantity, is better than your number or 27? I said your results are based on incomplete calculations, such as the formula you used to get your numbers to decimal places, lol. You're touting unfalsifiable claims that 27 is better because we should just trust whatever "cubic" formula you made. Come on.
You clearly underestimate the education of some people on the forums. Believe it or not, there are faculties of education higher than undergrad econ.
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On February 11 2011 07:01 natewOw wrote:Show nested quote +Btw, the cost is actually wrong it's 50 minerals for the worker +100/8 = 12.5 for the supply. The real cost is 62.5 pr worker. Please explain, I don't understand how you come to this conclusion. I don't even know what your unit of measurement is.
You are not taking suply into consideration at all. The cost of your 12th worker ist actually not 50 minerals, its 50 minerals, +100 minerals for a supply depot, + pulling 1 worker from mining for the time the depot needs to be constructed (and the time he needs to get to the construction site *2).
This being just one of many flaws in your analysis, I doubt that it has any practical value whatsoever.
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To everyone wondering how this is applicable in a real game, I have thought of a valid scenario:
Consider the following situation:
You have two bases, your main and your natural. Between them you have 40 workers.
Your main becomes mined out, and so you transfer all your workers mining at the main to the natural, while also building a third base.
You now have 40 workers mining at the natural, but any amount of workers above 27 gets you nothing, and may even decrease your net income due to cluttering.
Your third base is still under construction.
Using the knowledge you have gained from this thread, you should take 13 of the 40 workers and commence distance mining with them, since having 40 workers on the same base essentially means that 13 workers are doing nothing.
As soon as the third base finishes, those 13 workers are already primed to start delivering minerals to that base, so you don't have to waste time transferring workers from the natural, 13 of which weren't doing anything anyway.
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You clearly underestimate the education of some people on the forums. Believe it or not, there are faculties of education higher than undergrad econ.
Please tell me where I have made any assumptions or implications about anybody's education.
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On February 11 2011 08:01 CherubDown wrote:Maybe I'm missing something here. My question is: Why the heck is there a diminishing return before the 8th worker here? Let me know if I'm stupid as I have no experience in economics.
It's not diminishing return, but I see why you're confused. Honestly, it's a terrible graph. The problem is that I use open office, rather than MS office, and I don't know how to use it too well. I didn't look at any numbers before 7 workers, so I put in zero as placeholders for everything before 7 workers.
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On February 11 2011 07:58 natewOw wrote:Read the post dude, I said that I didn't measure quantities 1-6, so I put in zero as placeholders.
Sigh. I did read your post. Perhaps you put it in there as placeholders, however in your model using marginal cost per minute, at time 0 your total cost = 0 because you get those first 6 SCVs for free. Marginal cost is the derivative of total cost with respect to quantity, so at time 0 in your model, MC=0 as well.
You can't just change the definition of marginal cost and then continue to use traditional cost analysis.
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If you are forced to stack workers from running dry, It either means you are getting fucked, or you suck and forgot to expand. In the former case, distance mining is likely to get all your workers killed. Latter case, you just suck and need to l2p
Still doesn't apply
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On February 11 2011 08:09 natewOw wrote:
To everyone wondering how this is applicable in a real game, I have thought of a valid scenario:
Consider the following situation:
You have two bases, your main and your natural. Between them you have 40 workers.
Your main becomes mined out, and so you transfer all your workers mining at the main to the natural, while also building a third base.
You now have 40 workers mining at the natural, but any amount of workers above 27 gets you nothing, and may even decrease your net income due to cluttering.
Your third base is still under construction.
Using the knowledge you have gained from this thread, you should take 13 of the 40 workers and commence distance mining with them, since having 40 workers on the same base essentially means that 13 workers are doing nothing.
As soon as the third base finishes, those 13 workers are already primed to start delivering minerals to that base, so you don't have to waste time transferring workers from the natural, 13 of which weren't doing anything anyway.
This makes a lot of assumptions that cannot be stated with what you found (to be fair) being:
1) When does cluttering actually occur? Other people have stated at 30, so then you would have 30 workers on minerals not 27. Since your 27 number only talks about when it costs more to produce then use, but when already produced then...
2) This also assumes that the minerals gained by workers 25,26 and 27 are great when mining from your natural then from long distance mining... something that would be map dependant and needed to be testing.
3) This also assumes that the 3rd is not being taken at a gold expansion where long distance mining on say Xel'Naga Caverns or LT as examples would yeild gold minerals from long distance mining most likely larger than the amount of minerals mined by workers 25,26 and 27 at a single base.
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I honestly can't tell if I'm being trolled. Quote the rest of my post(s) for the real meat. Including the guy above asking about the DR @ 8 workers.
This thread is going nowhere fast because the OP is refusing to acknowledge any of the real logical problems of his approach and results brought up in the thread, instead going for the colloquial attacks.
Clearly this thread is going nowhere.
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On February 10 2011 01:58 LaLuSh wrote: After 22 workers mining minerals there is actually a gain from sending your workers distance mining. At least if the expansion is at a similar distance to that of LT’s 12 o’clock position.
Maybe you should reread LaLuSh's thread. If you think 27 is the optimal number when you get more money from long distance mining after 22 then your entire premise is redundant.
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On February 11 2011 08:09 natewOw wrote:
To everyone wondering how this is applicable in a real game, I have thought of a valid scenario:
Consider the following situation:
You have two bases, your main and your natural. Between them you have 40 workers.
Your main becomes mined out, and so you transfer all your workers mining at the main to the natural, while also building a third base.
You now have 40 workers mining at the natural, but any amount of workers above 27 gets you nothing, and may even decrease your net income due to cluttering.
Your third base is still under construction.
Using the knowledge you have gained from this thread, you should take 13 of the 40 workers and commence distance mining with them, since having 40 workers on the same base essentially means that 13 workers are doing nothing.
As soon as the third base finishes, those 13 workers are already primed to start delivering minerals to that base, so you don't have to waste time transferring workers from the natural, 13 of which weren't doing anything anyway.
This doesn't have to be true. Depending on the distance between your second and your in-construction-third sendig more than 13 workers distance mining might (taking your numbers surely will) generate a greater income.
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On February 11 2011 08:15 Insanious wrote:Show nested quote +On February 11 2011 08:09 natewOw wrote:
To everyone wondering how this is applicable in a real game, I have thought of a valid scenario:
Consider the following situation:
You have two bases, your main and your natural. Between them you have 40 workers.
Your main becomes mined out, and so you transfer all your workers mining at the main to the natural, while also building a third base.
You now have 40 workers mining at the natural, but any amount of workers above 27 gets you nothing, and may even decrease your net income due to cluttering.
Your third base is still under construction.
Using the knowledge you have gained from this thread, you should take 13 of the 40 workers and commence distance mining with them, since having 40 workers on the same base essentially means that 13 workers are doing nothing.
As soon as the third base finishes, those 13 workers are already primed to start delivering minerals to that base, so you don't have to waste time transferring workers from the natural, 13 of which weren't doing anything anyway. This makes a lot of assumptions that cannot be stated with what you found (to be fair) being: 1) When does cluttering actually occur? Other people have stated at 30, so then you would have 30 workers on minerals not 27. Since your 27 number only talks about when it costs more to produce then use, but when already produced then... 2) This also assumes that the minerals gained by workers 25,26 and 27 are great when mining from your natural then from long distance mining... something that would be map dependant and needed to be testing. 3) This also assumes that the 3rd is not being taken at a gold expansion where long distance mining on say Xel'Naga Caverns or LT as examples would yeild gold minerals from long distance mining most likely larger than the amount of minerals mined by workers 25,26 and 27 at a single base.
1) Even if cluttering starts at 30, it doesn't change the fact that workers 28 and 29 are not affecting your revenue per minute positively. You can only receive a certain maximum amount of income per minute from one base, and I have stated from the beginning that this maximum is realized when you have 27 workers on one base. And actually, this number doesn't talk about when it costs more to produce than to use. It talks about what amount of workers gets you the most minerals per minute (which is using the workers).
2) No arguments. It may net you more minerals to distance mine with 20 rather than 13. But given the work I've done here, I show you that you should never have more than 27 on one base.
3) Same as 2.
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I did my undergrad in economics and got top of my class so I know the theory you are basing this on fairly well. I have tried applying economics to Starcraft as well but generally find that it is not too useful. Mainly because starcraft is much more complex than many economics model allow for. The big problems are opportunity cost, risk and time.
The real concerns when playing starcraft is staying alive, and making sure you stay alive in the future too. This is based on what you do but also what your opponent does so is very hard to model. So optimality of revenue is a pretty small factor when deciding how many workers to build. What people care about is whether they can build it, stay alive and benefit in the long run or cut workers now and kill the opponent before their economy kicks in.
I have played with a few models but didn't post the results up on TL because the findings are so blatantly obvious to a diamond level plus player.
the human mind is too good at making calculations about risk and return, it is hard wired into our intuition to help is survive!
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On February 11 2011 08:20 madmaekki wrote:Show nested quote +On February 11 2011 08:09 natewOw wrote:
To everyone wondering how this is applicable in a real game, I have thought of a valid scenario:
Consider the following situation:
You have two bases, your main and your natural. Between them you have 40 workers.
Your main becomes mined out, and so you transfer all your workers mining at the main to the natural, while also building a third base.
You now have 40 workers mining at the natural, but any amount of workers above 27 gets you nothing, and may even decrease your net income due to cluttering.
Your third base is still under construction.
Using the knowledge you have gained from this thread, you should take 13 of the 40 workers and commence distance mining with them, since having 40 workers on the same base essentially means that 13 workers are doing nothing.
As soon as the third base finishes, those 13 workers are already primed to start delivering minerals to that base, so you don't have to waste time transferring workers from the natural, 13 of which weren't doing anything anyway. This doesn't have to be true. Depending on the distance between your second and your in-construction-third sendig more than 13 workers distance mining might (taking your numbers surely will) generate a greater income.
It's possible, but the 27 gives you the lower bound. If you want to get the most minerals off ONE base, it's 27.
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On February 11 2011 08:25 plagiarisedwords wrote:
I did my undergrad in economics and got top of my class so I know the theory you are basing this on fairly well. I have tried applying economics to Starcraft as well but generally find that it is not too useful. Mainly because starcraft is much more complex than many economics model allow for. The big problems are opportunity cost, risk and time.
The real concerns when playing starcraft is staying alive, and making sure you stay alive in the future too. This is based on what you do but also what your opponent does so is very hard to model. So optimality of revenue is a pretty small factor when deciding how many workers to build. What people care about is whether they can build it, stay alive and benefit in the long run or cut workers now and kill the opponent before their economy kicks in.
I have played with a few models but didn't post the results up on TL because the findings are so blatantly obvious to a diamond level plus player.
the human mind is too good at making calculations about risk and return, it is hard wired into our intuition to help is survive!
I'm not trying to model "the optimum amount of workers to win the game," all I did was show that to get the most minerals off of one base, you need 27 workers, not the previously-thought 24.
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Not sure why people are so hostile, when all they'd usually do is lurk on the forums and never contribute anything of value; great work  . People need to quit nit-picking the 'practicality' of this information, since there are many things still to be discovered in SC2 and we've got to explore the whole scope of it.
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All you guys who are reading this and responding with "OMG 1 BASE? 2 BASE IS BETTER" are yet to realise your error, because it's obvious 3 BASE IS BETTER LOL!
In all seriousness tho, natewOw, excellent post :D I'm sure a lot of people who read it will find it informative and I hope that they take away with them an insight into basic economic principles which will allow them to explore the issue further in a constructive manner.
In regards to optimizing your 'workers to base' count, I believe the point of this thread is to determine the potential maximum income per base, Yes it is obvious that having more bases will give you more income, but in a scenario of, for example, you have 5 bases, how many workers would you need to get a maximum possible income from 5 bases? (yes I am aware there is a supply cap, but it's not prudent for helping us understand these craftonomic theories). So please refrain from posting with comments like "just build more bases and don't saturate".
The above situation is more easily understood when you consider the 'worker to base' count of a 1-base scenario. Its relevant to early game, it's relevant to lategame when the map is mined out, and mid-game will just be a multiple of 1-base marginal revenues (assuming even worker distribution) depending on your base count.
I guess my only hang-up with your theory is that you've made the assumption that the game isn't going to immediately end and I'd be interested in modelling what happens when you change that assumption. Regardless I'd like to thank you natewOw for such a great post and inspiring me to explore some of these concepts myself :D
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On February 11 2011 08:27 sl10 wrote:Not sure why people are so hostile, when all they'd usually do is lurk on the forums and never contribute anything of value; great work .
Honestly, people questioning the usefulness of this doesn't bother me. I never said I was introducing some earth-shattering fact that was going to change the shape of the game.
It's the people trying to tell me that 24 workers on one base gets you more income-per-minute than does 27, despite me showing empirical evidence that this is not the case.
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If it would take 20 minutes or so for the extra 3 workers to pay off their cost, wouldn't the patches be mined out by then?
And 27 workers does generate more income than 24, common sense, but is that all that you're trying to prove?
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On February 11 2011 08:29 natewOw wrote:Show nested quote +On February 11 2011 08:27 sl10 wrote:Not sure why people are so hostile, when all they'd usually do is lurk on the forums and never contribute anything of value; great work . Honestly, people questioning the usefulness of this doesn't bother me. I never said I was introducing some earth-shattering fact that was going to change the shape of the game. It's the people trying to tell me that 24 workers on one base gets you more income-per-minute than does 27, despite me showing empirical evidence that this is not the case.
How is it empirical? Your data is from a formula you don't even disclose, not actual in-game observations.
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Dear natewOw,
First, thanks for undertaking this analysis--I am a bit of an economics geek myself, and I have been enjoying LaLush's (and now your) post and comments a lot. I think good analysis of this type can advance our understanding of the game considerably.
Some specific thoughts on your analysis:
-you've chosen to define MC and MR in terms of minerals/time. This is obviously fine. However, your argument that MC->0 as T->inf introduces a flaw in your analysis because given that the number of minerals available on any map is fixed, MR->0 as T->inf as well (i.e., if you have an unbounded amount of mining time, you can mine out the map with only one worker, or with just your original workers, or however you want to look at it). To the extent that this isn't really a practical consideration, then neither is the assumption that we should be calculating the limit of MC as T->inf. Why not just step back and say you've calculated the minimum worker count that produces the maximum revenue flow for a single base? (Note: this may not even be exactly true given LaLush's interesting finding that MR from distance mining may exceed MR from one-base at around 22 workers--I'd love to see this tested more.)
-you've chosen to estimate MR as a cubic function. LaLush and his predecessor chose to use an empirical estimate of MR (or related values). I'm not convinced yours is the preferable approach, particularly considering that your estimated function is concave and turns negative in the neighborhood of 20-30 workers. My prior on this is that it must be that MR->0 as workers->inf, and that MR cannot turn negative, or at least very negative. You need your estimated function to be most accurate in this range to prove your point, but its accuracy seems to be questionable in just that range. You write "...I generated predicted values of income per minute as a cubic function of the number of workers currently mining. Don't get too hung up on the method..." I'd actually love to hear the method you used--did you base it on some empirics you did, or that you got from another source?
I hope you take the feedback you're getting in stride, and that it prompts more of this brand of analysis from you and the rest of the community. I hope to add something in this vein at some point when I feel I have the time.
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On February 11 2011 07:09 natewOw wrote:Show nested quote +I am wondering if you have the right definition for "Optimal Saturation". Here, you define optimal saturation as the maximum rate of mineral collection. You got it right, optimal saturation by my definition is the most minerals that can be collected per minute. I said nothing about build orders.
Nice response, but I wanted to ask you an idea about another economic concept. Your original argument says that the marginal cost goes to zero as T goes to infinity, and I thought it was a clever argument.
But what about the opportunity cost of the minerals to used to build the probe? That is, when you chose to build, you forgo the benefits of building a building or attacking unit, and those forgone benefits are a cost.
So how can the marginal cost of a probe go to zero, when it seems that the opportunity costs always exist (regardless of time)?
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On February 11 2011 08:34 mucker wrote:Show nested quote +On February 11 2011 08:29 natewOw wrote:On February 11 2011 08:27 sl10 wrote:Not sure why people are so hostile, when all they'd usually do is lurk on the forums and never contribute anything of value; great work . Honestly, people questioning the usefulness of this doesn't bother me. I never said I was introducing some earth-shattering fact that was going to change the shape of the game. It's the people trying to tell me that 24 workers on one base gets you more income-per-minute than does 27, despite me showing empirical evidence that this is not the case. How is it empirical? Your data is from a formula you don't even disclose, not actual in-game observations.
I should have made this clearer in the post. It is from actual in-game observations. There's really no "formula", I just estimate more precise measurements of the marginal revenue increments, since blizzard's income meter only goes in increments of 20.
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On February 11 2011 08:37 natewOw wrote:Show nested quote +On February 11 2011 08:34 mucker wrote:On February 11 2011 08:29 natewOw wrote:On February 11 2011 08:27 sl10 wrote:Not sure why people are so hostile, when all they'd usually do is lurk on the forums and never contribute anything of value; great work . Honestly, people questioning the usefulness of this doesn't bother me. I never said I was introducing some earth-shattering fact that was going to change the shape of the game. It's the people trying to tell me that 24 workers on one base gets you more income-per-minute than does 27, despite me showing empirical evidence that this is not the case. How is it empirical? Your data is from a formula you don't even disclose, not actual in-game observations. I should have made this clearer in the post. It is from actual in-game observations. There's really no "formula", I just estimate more precise measurements of the marginal revenue increments, since blizzard's income meter only goes in increments of 20.
So there is no magical "cubic" formula, and to put it simply, you pause the game and jot down on a piece of paper the minerals you have at a given time?
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On February 11 2011 08:37 SolonTLG wrote:Show nested quote +On February 11 2011 07:09 natewOw wrote:I am wondering if you have the right definition for "Optimal Saturation". Here, you define optimal saturation as the maximum rate of mineral collection. You got it right, optimal saturation by my definition is the most minerals that can be collected per minute. I said nothing about build orders. Nice response, but I wanted to ask you an idea about another economic concept. Your original argument says that the marginal cost goes to zero as T goes to infinity, and I thought it was a clever argument. But what about the opportunity cost of the minerals to used to build the probe? That is, when you chose to build, you forgo the benefits of building a building or attacking unit, and those forgone benefits are a cost. So how can the marginal cost of a probe go to zero, when it seems that the opportunity costs always exist (regardless of time)?
I actually addressed this earlier, but even if you include opportunity cost within marginal cost, the marginal cost still converges to zero, because you are dividing MC by time, T. Thus, MC could include the cost of my dry cleaning, and it would still go to zero as the game time gets infinitely larger.
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On February 11 2011 08:39 Nagano wrote:Show nested quote +On February 11 2011 08:37 natewOw wrote:On February 11 2011 08:34 mucker wrote:On February 11 2011 08:29 natewOw wrote:On February 11 2011 08:27 sl10 wrote:Not sure why people are so hostile, when all they'd usually do is lurk on the forums and never contribute anything of value; great work . Honestly, people questioning the usefulness of this doesn't bother me. I never said I was introducing some earth-shattering fact that was going to change the shape of the game. It's the people trying to tell me that 24 workers on one base gets you more income-per-minute than does 27, despite me showing empirical evidence that this is not the case. How is it empirical? Your data is from a formula you don't even disclose, not actual in-game observations. I should have made this clearer in the post. It is from actual in-game observations. There's really no "formula", I just estimate more precise measurements of the marginal revenue increments, since blizzard's income meter only goes in increments of 20. So there is no magical "cubic" formula, and to put it simply, you pause the game and jot down on a piece of paper the minerals you have at a given time?
Actually I jotted down the replay's estimation of the income-per-minute at X number of workers. Like I said, this only goes in 20 mineral increments and is no doubt an estimation, so I estimated more precise by making income-per-minute a function of workers, workers^2, and workers^3.
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So why is there a diminishing return in MR between 7 and 8 workers if you assume your MC is 0?
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United States7483 Posts
I understand your reasoning and logic behind your arguments, and why MR is measured by time, but I have to take issue with your units here, the marginal cost of an additional worker being 0 does not lead to optimal play.
I agree that you have shown that more than 24 workers on one base yields more minerals than exactly 24, but the units you are using makes it impossible to accurately determine if it's worth 'enough' more minerals per minute to make those extra workers while staying on one base. It's difficult to work around this however, with the way the game functions by using economic models.
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You have two bases, It's possible, but the 27 gives you the lower bound. If you want to get the most minerals off ONE base, it's 27.
You could show that with 97% fewer graphs. Why does introducing marginal revenue and so on show anything more than the statement '27 workers gives you maximum income off one base'?
And the distance mining calculation was done better by Lalush.
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On February 11 2011 08:09 natewOw wrote:
You now have 40 workers mining at the natural, but any amount of workers above 27 gets you nothing, and may even decrease your net income due to cluttering.
what do you mean by cluttering? There is no such thing as cluttering when workers mine cause they pass through eveything, including other workers.
just curious
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OP spends an hour writing up some fancy econ post to show that 27 > 24 miners on 1 base and when he is confronted with the fact that this is useless in a real game (unlike Lalush's excellent thread on this topic), he then fills this thread with stubborn passive-aggressive bait posts (aka trolling). I don't see this thread lasting very long.
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The optimal number of workers per base is 2 per mineral patch. When above 2 workers per patch you get diminishing returns per worker. I mean why would a player not want to have a worker pay for itself asap. If you have more then 2 workers per patch it takes longer for the workers to pay for themselves. I don't understand why people think it is anything else. The only reason to go above 2 per patch is if you can not safely expand(granted this happens a lot).
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There is a HUGE flaw in the numbers that I think most people have just simply overlooked.
In the OP: "This is saying that the limit of the marginal cost function, as time gets infinitely greater, is zero. Thus, the expected marginal cost of producing an additional probe is zero."
If MC is zero, so then why is there a diminishing return on MR when there are 16 or less workers (assuming 8 mineral patches)? Does this make no sense to anyone else?
This would mean that the graph above would be completely false up to 16 workers. Am I missing something?
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On February 11 2011 08:54 SirazTV wrote:
The optimal number of workers per base is 2 per mineral patch. When above 2 workers per patch you get diminishing returns per worker. I mean why would a player not want to have a worker pay for itself asap. If you have more then 2 workers per patch it takes longer for the workers to pay for themselves. I don't understand why people think it is anything else. The only reason to go above 2 per patch is if you can not safely expand(granted this happens a lot).
Why do people buy stocks now that won't give them any returns for a year or more? It's called an investment. Less minerals now = more later.
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Out of all the responses, that's the one you quote? 
Reread please.
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On February 11 2011 08:59 Nagano wrote:There is a HUGE flaw in the numbers that I think most people have just simply overlooked. In the OP: "This is saying that the limit of the marginal cost function, as time gets infinitely greater, is zero. Thus, the expected marginal cost of producing an additional probe is zero." If MC is zero, so then why is there a diminishing return on MR when there are 16 or less workers (assuming 8 mineral patches)? Does this make no sense to anyone else? This would mean that the graph above would be completely false up to 16 workers. Am I missing something?
It's a graph of ESTIMATES, not true values. It's meant to show the shape of the returns function, not the specific values themselves.
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On February 11 2011 08:51 Soulish wrote:Show nested quote +On February 11 2011 08:09 natewOw wrote:
You now have 40 workers mining at the natural, but any amount of workers above 27 gets you nothing, and may even decrease your net income due to cluttering.
what do you mean by cluttering? There is no such thing as cluttering when workers mine cause they pass through eveything, including other workers. just curious
Even if there's no cluttering (which I agree, I don't think there is, but I've heard that other people have suggested that there is), it doesn't detract at all from what I said.
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On February 11 2011 09:02 natewOw wrote:Show nested quote +On February 11 2011 08:59 Nagano wrote:There is a HUGE flaw in the numbers that I think most people have just simply overlooked. In the OP: "This is saying that the limit of the marginal cost function, as time gets infinitely greater, is zero. Thus, the expected marginal cost of producing an additional probe is zero." If MC is zero, so then why is there a diminishing return on MR when there are 16 or less workers (assuming 8 mineral patches)? Does this make no sense to anyone else? This would mean that the graph above would be completely false up to 16 workers. Am I missing something? It's a graph of ESTIMATES, not true values. It's meant to show the shape of the returns function, not the specific values themselves.
You are COMPLETELY missing the point here. There should be ZERO DR on MR at any point before 16 workers. Half the data is wrong, and therefore the entire graph should look completely different.
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Well the thing is, tons of players in this thread are right.
The OP is 100% correct that in a game of infinite length, 27 workers > 24 workers.
However, as has been pointed out, it takes over 20 minutes for that last worker to pay for himself, while the opportunity cost of building those additional workers early on stays the same.
And appart from the opportunity cost, the other issue you have, is that minerals will actually mine out rather fast from a single base with optimal saturation.
So while its true that 27 SCVs would be the optimal number to stop at in a game of infinite time where mineral patches never mine out, and you stay on a single base for the whole game, unfortunately, it is quite irrelevant for the actual gameplay itself, since in an actual game, you either want to expand relatively soon, at which point building extra workers past 27 is still a good idea, or you want to 1 base all-in relatively soon, at which point the opportunity cost for those extra 3 workers is not worth it.
However though, I believe that the OP's formula probably can be used quite well, as long as it is solved for a more reasonable number than T=infinity.
I tried it out, and it seems to take 17 minutes to fully mine out a base, anjd 15 minutes to fully mine out a base with mules.
So perhaps the OP, or someone else that is good at maths could retake the base formula, and solve it for T=15 minutes, instead of for T= infinity, and we could check the results?
Perhaps we will get some new information, the number might be smaller
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On February 11 2011 09:03 Nagano wrote:Show nested quote +On February 11 2011 09:02 natewOw wrote:On February 11 2011 08:59 Nagano wrote:There is a HUGE flaw in the numbers that I think most people have just simply overlooked. In the OP: "This is saying that the limit of the marginal cost function, as time gets infinitely greater, is zero. Thus, the expected marginal cost of producing an additional probe is zero." If MC is zero, so then why is there a diminishing return on MR when there are 16 or less workers (assuming 8 mineral patches)? Does this make no sense to anyone else? This would mean that the graph above would be completely false up to 16 workers. Am I missing something? It's a graph of ESTIMATES, not true values. It's meant to show the shape of the returns function, not the specific values themselves. You are COMPLETELY missing the point here. There should be ZERO DR on MR at any point before 16 workers. Half the data is wrong, and therefore the entire graph should look completely different.
The data isn't wrong. I could have posted the raw numbers, but that graph would have looked very spiky and strange and wouldn't have made sense to people.
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Since we do not know how long a game will last, we want to define the marginal cost function as the game time gets infinitely greater:
lim(50/T) as T → ∞ = 0
This is saying that the limit of the marginal cost function, as time gets infinitely greater, is zero. Thus, the expected marginal cost of producing an additional probe is zero.
no. the expected value of the marginal cost is the integral over all feasible values for T, the integrand is the marginal cost function as a function of T, multiplied with a probability density function which, roughly speaking, assigns each value of T a certain probability. that the marginal cost function converges to zero for T -> infinity does NOT imply that said expected value is zero. a very simple example: at any given point in time during a game, the remaining duration of the game can theoretically vary between 0 and 10000000 minutes. but in reality, it is over 9000 times more likely for the game to end in 10 minutes than to end in 200 minutes. the probability for high values of T approaches zero at a very fast rate, fast enough to make the zero marginal cost for these very large T-values irrelevant for the overall integral.
TLDR: that the marginal cost function converges to zero for T -> infinity does NOT imply that the expected marginal cost of producing one more worker is zero. therefore, your analysis completely fails on a very basic mathematical level. sorry if i sound harsh now, but i´d say back to calc and stats 101.
if an additional worker does not increase the mining rate at all (ie we already have 24+ workers on minerals per base), it is simply useless to build more workers if one does not plan to expand in the near or intermediate future or is at immediate risk of losing workers to harass. in reality though, these conditions will almost always be fullfilled so that it is almost always advisable to continue worker production unless one has very specific and good reasons not to.
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On February 11 2011 09:08 natewOw wrote:Show nested quote +On February 11 2011 09:03 Nagano wrote:On February 11 2011 09:02 natewOw wrote:On February 11 2011 08:59 Nagano wrote:There is a HUGE flaw in the numbers that I think most people have just simply overlooked. In the OP: "This is saying that the limit of the marginal cost function, as time gets infinitely greater, is zero. Thus, the expected marginal cost of producing an additional probe is zero." If MC is zero, so then why is there a diminishing return on MR when there are 16 or less workers (assuming 8 mineral patches)? Does this make no sense to anyone else? This would mean that the graph above would be completely false up to 16 workers. Am I missing something? It's a graph of ESTIMATES, not true values. It's meant to show the shape of the returns function, not the specific values themselves. You are COMPLETELY missing the point here. There should be ZERO DR on MR at any point before 16 workers. Half the data is wrong, and therefore the entire graph should look completely different. The data isn't wrong. I could have posted the raw numbers, but that graph would have looked very spiky and strange and wouldn't have made sense to people.
Generally if a graph is very spiky and strange you have inconclusive data...
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You aren't listening. There are reductions in MR at every point before 16 workers. There should be NO reduction in MR before 16 workers assuming 8 mineral patches if you assume zero MC.. At best the first half of that graph (up to 16) should be flat. Even if you were smoothing it for aesthetic purposes, there should be no negative slope up to 16 workers! Therefore how can you trust ANY of the numbers, including your conclusion of 27 workers, which is derived from your graph?
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Workers cost 62.5 minerals because each supply building (or overlord) costs 100 minerals and provides 8 supply.
In other words, your first four workers cost 50 minerals each as Protoss and Zerg because you start with 6/10 supply. For Terran the cost is pretty much the same because, to get more supply, you have one worker that does nothing for a little bit more than the time it takes to create an additional worker (so the +1 supply in the early game matters for little)
So, after the 10th worker, each worker costs 62.5 minerals until an expansion goes up and provides you with more supply. This is clearest with Zerg, since hatcheries only provide 2 supply each.
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an additional aspect which the analysis in the OP does not account for is the opportunity cost of supply: additional workers take up one supply which cant be used for army. the closer a player gets to the supply cap of 200, the higher the opportunity cost of taking up a supply spot by a nonfighting unit. factoring in the increasing opportunity cost of supply, the marginal cost of additional workers can never be zero and thus the increase in mining rate needs to be ABOVE zero to justify the production of this worker, which obviously means that from an economic point of view it is no good idea to supersaturate.
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On February 11 2011 09:11 Nagano wrote:
You aren't listening. There are reductions in MR at every point before 16 workers. There should be NO reduction in MR before 16 workers assuming 8 mineral patches if you assume zero MC.. At best the first half of that graph (up to 16) should be flat. Even if you were smoothing it for aesthetic purposes, there should be no negative slope up to 16 workers! Therefore how can you trust ANY of the numbers, including your conclusion of 27 workers, which is derived from your graph?
You don't have to trust me. Run it yourself, you will see that 27 workers gets you higher income-per-minute than 24.
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care to show more info on the cubic in the op? Also after 27 the income/minute wont go down, the negative part of your cubic does not kick in even if it tends to 0
Income tends to a maximum rate after which its a waste of the 50 minerals to make a worker once 100% saturation is hit
Theres plent of graphs where it shows worker income increasing untill 28-32 before flatlining, but in a real game you wont go past 20-24, so 3/ patch is a good rule of thumb limit
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[B]that the marginal cost function converges to zero for T -> infinity does NOT imply that said expected value is zero. a very simple example: at any given point in time during a game, the remaining duration of the game can theoretically vary between 0 and 10000000 minutes. but in reality, it is over 9000 times more likely for the game to end in 10 minutes than to end in 200 minutes. the probability for high values of T approaches zero at a very fast rate, fast enough to make the zero marginal cost for these very large T-values irrelevant for the overall integral.
This is my concern as well. I don't believe you can draw the conclusion that a 27th probe will always be worthwhile since it's based on the assumption of a game of infinite length (as others have pointed out). I'm not sure you can actually draw any conclusions in this area since there's no exact method of estimating how long a game will last at any point before the final moment of the game.
However, that's not to say the numbers have no value. I think a more meaningful statistic would be to determine that amount of time it takes for each probe to pay itself off. That value has a number of practical applications, especially for all-in builds that win or die at a specific moment in time.
For example, if you have an all-in build that attacks at 6 minutes, is it worth build a probe at 5 minutes that takes 2.5 minutes to pay itself off? Probably not...you'd be better off spending the 50 minerals and 1 supply on something else.
To the 27 vs 24 probes argument, I can see some times when this would be useful (assuming its accurate). There are maps where it is challenging to take and hold a 3rd base. It is useful information to know that the 25, 26, and 27th probes on each base generate additional income, especially if I have an excess of minerals that I can't spend for some reason. Although again, there is a cost associated with this. Is it better to build the additional probe or to build a gateway that will be used <100% of the time? I'm not sure how to answer that question.
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On February 11 2011 09:19 natewOw wrote:Show nested quote +On February 11 2011 09:11 Nagano wrote:
You aren't listening. There are reductions in MR at every point before 16 workers. There should be NO reduction in MR before 16 workers assuming 8 mineral patches if you assume zero MC.. At best the first half of that graph (up to 16) should be flat. Even if you were smoothing it for aesthetic purposes, there should be no negative slope up to 16 workers! Therefore how can you trust ANY of the numbers, including your conclusion of 27 workers, which is derived from your graph? You don't have to trust me. Run it yourself, you will see that 27 workers gets you higher income-per-minute than 24.
Your graph is 100% wrong, it should have no negative slope before point 16. Black Gun just explained to you the enormous pitfalls in your mathematical logic.
Your method involved running a game and observing the income tab. You fabricated the rest of the information used to generate the downward sloping graph, including faking your "cubic" function and results, because the real graph of MR would look nothing remotely close to the one you created. You assumed infinite mineral values per patch and an MC of zero. The entire OP was a fabrication attempting to hide your basic assumptions and mathematical ability behind a guise of a one month depth in econ 1 knowledge.
Someone close this thread, please.
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On February 11 2011 07:20 natewOw wrote:
It's just fact? Where are you getting this "fact" from? Because I ran a simulation comparing 24 workers to 27, and 27 was netting me more minerals. Can you offer any evidence to back up your fact?
Simulation?
What happens in the game? I don't know either way, but surely it's better to look?
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@natewow: The question you set out to ask doesn't require any of the economics theory. If you choose discard all the context of an actual game of starcraft, then time doesn't factor in. If you are trying to maximize the minerals you have to spend, you should never build any workers beyond the six you start with, because the minerals on the map are finite. If you are trying to maximize the rate of mineral income at a single base, that is an academic exercise best solved by direct measurement (if you doubt the logic of 24 workers), specific to each expansion on each map, because the mineral patch placements differ. (I guess there will be some exact duplicates across the maps.) You are optimizing something arbitrary, which is fine. You set out to find the least amount of workers that gives you maximum mining rate at a single base. You stated something about the MR after 20 minutes... an expansion's minerals don't last that long. The idea of considering what happens as T increases is simply an inaccurate portrayal of the situation being modeled. This is not to say it isn't an interesting abstract perspective.
The most accurate way to measure the income rate at a given number of workers would be to see how much you mine in a set time period, say 5 minutes. Minerals mined / 5 minutes. I assume this would incline until plateauing at 24, unless my understanding of the worker AI isn't correct. From what I see, the workers will bounce quite a bit until eventually settling, where they pair or triple up. If you have every patch tripled up, every patch is being mined at every possible moment. There is no way an extra worker could access further mining time. This might be wrong, in which case the optimum worker count is near to 22-26.
@everyone else: Yes, we know. If you don't agree with OP, don't post. As many have pointed out, it makes no difference anyway. I hate watching intractable arguments over inconsequential viewpoints.
...Sorry for the outburst.
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I'm sorry I'm not a math major but I did my best to double check my numbers, feel free to check them yourself. Assuming you don't chrono boost the extra 3 (why would you) my calculations using your income numbers from the OP, put you mining out at ~16:15 with 0 chrono boost used on probes and no gas being mined. If I had built 24 probes in the same way, I would be mined out in ~16:28 with a disadvantage of 30 minerals (counting the 150 spent on the probes) at the 16:15 mark.
Now I hate numbers like these because there would never be a situation where your probes NEVER leave the mineral line, don't chrono probes, don't scout and you don't mine any gas in the first 16 min of a game. however doing any of these things will make the margin even smaller, as you will be closer to mined out when you get to the magic 27 probes on minerals.
When being held in my base I do advocate making extra havesters for the event in which the pressure is repelled, you have them ready to transfer as many people have already stated, however I can see no merit in having 27 probes per mineral line on 2 bases on say close position metal or somewhere where a 3rd is difficult.
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Just read the OP and most of the responses. First of all, i have to say i feel bad for the OP because so many people are being simply rude/passive aggressive and poiting out that 3x8=24 which somehow seems to be supposed to prove something.
Regarding the OP i am actually very bad at understanding economics and most math beyond what statistics for psychology education covers (which is basically using a computer) so my request and feedback would be to ask you to explain some of the terms a bit more specifically for those of us who are clueless when it comes to economics.
Otherwise great thread and dont get discouraged or mad at other people even if they are being condescending. It's easy to fall into that trap.
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On February 11 2011 09:28 Dragar wrote:Show nested quote +On February 11 2011 07:20 natewOw wrote:
It's just fact? Where are you getting this "fact" from? Because I ran a simulation comparing 24 workers to 27, and 27 was netting me more minerals. Can you offer any evidence to back up your fact? Simulation? What happens in the game? I don't know either way, but surely it's better to look?
Just to put a nail in this one, I decided to actually test the minerals mined over a 1 minute period at 24, 25, 26, and 27 probes. I did the test on Steppes of War and allowed the probes to settle over a 2 minute period before taking any numbers. I also didn't test any number of probes less than 24 as I believe that range has been tested before.
24 Probes - Top starting position
Minute 1 = 785 minerals
Minute 2 = 790 minerals
25 Probes - Bottom starting position
Minute 1 = 810 minerals
Minute 2 = 785 minerals
26 Probes - bottom natural
Minute 1 = 815 minerals
Minute 2 = 810 minerals
27 Probes - top natural
Minute 1 = 815 minerals
Minute 2 = 810 minerals
Given these numbers, I don't think it's safe to say that 27 probes produces any more minerals than 24 probes. The variance in mining rates can easily be accounted for by the time I pressed pause and wrote down the numbers. If I had waited a fraction of a second longer or paused a fraction of a second sooner, the numbers would undoubtedly be different as probes are constantly dropping off minerals at the Nexus.
So yeah...sorry to say but I think 24 probes is the point of maximum saturation.
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There are way too many variables to even bother calculating the exact number of workers for optimal output. When you factor in time away from mineral gathering, such as constructing buildings (or sac'ing yourself to make a bldg, read: drones), scouting, losing workers to harassing, transferring to for efficient expo'ing, proxy pylons, making bunkers, repairing mech, it's simple to reach the conclusion that in 99% of occurances (with the 1% being a planned 1 base all-in without getting harrassed,scouting, or using the workers to attack at the end), more workers are better, until you can blatantly see that you're oversaturated) regardless of what economic principles you want to apply. Factor in the opportunity cost of building workers instead of your army, and what you're left with is a strategy game..not an economics game.
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On February 11 2011 10:26 supernovice007 wrote:Show nested quote +On February 11 2011 09:28 Dragar wrote:On February 11 2011 07:20 natewOw wrote:
It's just fact? Where are you getting this "fact" from? Because I ran a simulation comparing 24 workers to 27, and 27 was netting me more minerals. Can you offer any evidence to back up your fact? Simulation? What happens in the game? I don't know either way, but surely it's better to look? Just to put a nail in this one, I decided to actually test the minerals mined over a 1 minute period at 24, 25, 26, and 27 probes. I did the test on Steppes of War and allowed the probes to settle over a 2 minute period before taking any numbers. I also didn't test any number of probes less than 24 as I believe that range has been tested before. 24 Probes - Top starting position Minute 1 = 785 minerals Minute 2 = 790 minerals 25 Probes - Bottom starting position Minute 1 = 810 minerals Minute 2 = 785 minerals 26 Probes - bottom natural Minute 1 = 815 minerals Minute 2 = 810 minerals 27 Probes - top natural Minute 1 = 815 minerals Minute 2 = 810 minerals Given these numbers, I don't think it's safe to say that 27 probes produces any more minerals than 24 probes. The variance in mining rates can easily be accounted for by the time I pressed pause and wrote down the numbers. If I had waited a fraction of a second longer or paused a fraction of a second sooner, the numbers would undoubtedly be different as probes are constantly dropping off minerals at the Nexus. So yeah...sorry to say but I think 24 probes is the point of maximum saturation.
I think you just pointed to the opposite of your conlusion. You have an overall increase which seems to be fairly small but linear from 24-27 workers. Then you attribute that variance purely to error (which is random) and then you jump to the conclusion that error variance explaining your results means that 24 is optimal? I dont understand your logic. Plus if you are going to show me that your data is statistically non-significant i will point out that it is really not very applicable here for various reasons.
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If I may offer a suggestion: I believe you are not taking in to account the fact that all scvs cannot be mining simultaneously. This theory makes sense assuming an unlimited amount of patches to mine from and to account for this I think you are assuming a base expansion. Is that correct?
Also, mineral patches eventually run out. Maybe 27 workers should be the optimal level of saturation for one base but only 8 workers can mine at once. Perhaps in this situation all workers are not getting their optimum level of mining time because of the latency created when they have to wait to gain access to a mineral patch.
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MC is wrong and so is MR.
MC = 50
MR = the expected total minerals harvested by the end of the game from having that 1 extra drone.
At best MR is zero when all mineral patches are fully saturated and an extra drone just does nothing. It can never be negative.
By this method, I think you'll get the 'conventional' result that we should fully saturate our mineral patches. Most games will last long enough to be worth that.
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to expand on my post from the previous page, with a income advantage of only 20 minerals per min, it will take you 7:30 game time to pay for the 3 probes. as I mentioned earlier unless you chrono boost while building #25, 26 or 27 (which would be impractical to say the least) you will have a mineal lead over 24 probes for a maximum of 2:48 game time, at which point you will be completely mined out. Again as I said in my previous post these are perfect world no gas no downtime numbers, doing ANYTHING you would do in a normal game will result in fewer minerals being left when you get to 27 probes on minerals, making the margin even worse.
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5003 Posts
@OP: the reason why you're getting such bad responses is because you convoluted the entire thing in unneeded econ terms. Not that there's actually econ in this -- there's zero economics involved in your model which is why it's just confusing people
if you just stated it for what it is ("At what point do you stop getting positive returns for adding a probe") then no one would have an issue.
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I liked to read the OP. The usefulness in regards to actually playing the game can be discussed, but that doesn't make it irrelevant. Some people might enjoy relating StarCraft to economic theory and discuss it.
If people want to nitpick whether 24-27 (or another number) workers is optimal saturation, they should at least carefully consider the things which comes into play. I haven't, but stuff such as mineral formations on different maps comes to mind. That certainly plays a minor role when nitpicking.
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On February 11 2011 10:48 Genovi wrote:
I think you just pointed to the opposite of your conlusion. You have an overall increase which seems to be fairly small but linear from 24-27 workers. Then you attribute that variance purely to error (which is random) and then you jump to the conclusion that error variance explaining your results means that 24 is optimal? I dont understand your logic. Plus if you are going to show me that your data is statistically non-significant i will point out that it is really not very applicable here for various reasons.
How is it linear when 26 and 27 are exactly the same? Even 25 probes shows a range of 785 to 810 versus a range of 810 to 815 for 26 and 27.
You could argue that there is a small increase from 24 to 25 but my feeling is that this is attributable to variations in timing rather than some increase in efficiency. Minerals increase in increments of 5 only when the probe drops off the minerals. A probe that is a one pixel from dropping off minerals accounts for zero minerals exactly as a probe that is half done or a quarter done with his mining path. We have no way of accurately determining partial values so we have to accept some flex in the mining numbers.
In other words, if I had waited another half second, the numbers would be different.
My conclusion that the differences are insignificant is based on the range of values found even within a single probe count and small sample size.
I'll concede that I might be jumping to conclusions when saying that there is no difference between 24 and 25 since I'd need to do alot more iterations to rule out the variance but I'm comfortable saying there is no difference between 25, 26, and 27.
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Basically, the OP can be summed up like that:
Assuming that every probe mines forever and minerals never run out even the slightest increase in income (eg 0.001 minerals per minute) will result in a probe paying for itself.
All the ecobabble is completely irrelevant given the basic assumption.
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I have to criticize this analysis for the sole reason that the OP states at the end of the post that the results show how to get the most minerals from one base. This is an example of a statistical analysis that does not take game mechanics into account.
Getting the most minerals from one base (in the shortest amount of time) cannot be achieved by constantly making workers. I don't have the exact numbers handy, but as a few people have previously shown getting more than 25/26 workers does not increase your minerals per minute.
More importantly oversaturating the mineral line will increase the amount of time it takes to completely mine out one base. Even though the workers should mine out a base above saturation in the same amount of time, it seems that the increased frequency of "mineral bouncing" that happens when a probe reaches a saturated mineral patch actually decreases mining efficiency. In numbers above 28, oversaturation can increase the amount of time it takes to fully mine one base by 20-30 seconds (which is pretty significant if you are staying on one base).
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Thanks for taking the time supernovice007.
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Hah, in economics when you have uncertainty about a variable you don't just assume its infinity and call that taking an expectation!
Thats like saying, I don't know what the stock market will do tomorrow so we have to assume its going to go to infinity, if you invest a penny today you'll be a millionaire tomorrow.
When you have uncertainty you have to actually take a MATHEMATICAL EXPECTATION by integrating over a distribution of possible game lengths.
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hm, im a little bit confused, cause i dont see why you would need all that economics stuff to answer your question. maybe thats because i missunderstood the question you tried to answer, so correct me if this is wrong:
you are trying to find the number of workers needed to maximize your income from one base(minerals mined from 8 mineral patches per time)
assuming adding workers increases income until theres a number of x workers after which adding more workers will not increase your income(respectively income stays constant or decreases because of workers getting into the way of each other), "all you have to do" is figure out the income for 1 worker, 2 workers, and so on until you dont see an increase in income.
i think from a theoretical point of view(assuming you know the game will go on long enough and there are enough minerals) its quite intuitive that up until that number x its "worthwile" to produce workers, cause all of them add to your income, thus eventually will "pay off" (after y minutes the additional income created through that worker will be higher than the cost of that worker).
i think you tried to explain that fact with all that "economics stuff", right?
so whether x=24 or x=27 or x=something else basically comes down to the numbers for the income you used. In your first post you wrote: "I generated predicted values of income per minute as a cubic function of the number of workers currently mining", which sounds to me like you used some kind of function to simulate those income numbers. if so you should show us this function and tell us why the values generated by it are the correct numbers for the income.
later on you said something like you used ingame observations to get those numbers. in this case i doubt your observations reflect the actual income rates, because there are sources of measuring error, for example:
1. How did you make sure you were measuring the income over the exact same time( and not 60.0 vs 60.5 seconds or something?)
2. How did you make sure you did not start measuring the income for 24 workers 1 millisecond AFTER 8 workers brought in a mineral patch and the income for 25 workers 1 millisecond BEFORE 8 workers brought in a mineral patch?
these two things alone might falsify your numbers enough to bring you to wrong conclusions.
on another topic: i unterstand that you approached this from a theoretical point of view, but if you want to find applications in actual games, you probably have to consider the maximal number of minerals that can be mined. for example, the whole map is mined out except for one base. according to your numbers getting a 27th worker on minerals might be ok, because he pays of after 20 minutes, but in reality the last base will be mined out after 15 minutes, so you are actually losing minerals with the 27th worker
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Your argument that the marginal cost is 0 is obviously incorrect, because games do not last infinitely long. If you know you're going to all-in in 10 seconds, then building another probe will not net you increased income.
Also, what evidence is there that increasing your worker count from 24 to 27 actually increases your income? Your graph says it, but your graph already assumes that having more miners past 24 will increase your income. It doesn't seem to consider max saturation.
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If nothing else, OP is doing a great job at reinforcing the stereotype of economists among engineers - takes some data, applies numbingly simple analysis and covers it all in unnecessary terminology, and then fights off all and any critic to the ill-fitting model with beak and claws.
Why does the marginal revenue go down from 7 to 8? The increase in income should be 100% linear, since the 8th worker is using a patch previously unused. Why does it go down between 9 and 16? Each of those workers modify a patch worked on by 1 worker into a patch worked on by 2 workers - the increase in income should be the same for each of these. Why? Because you've applied a poor model to the data - it even goes negative at a point, something you defend by pointing to cluttering, rather than acknowledging the limitations of your model!
Also, letting time go to infinity thus saying that the expected marginal cost for adding a worker is zero is very strange - why did you even bring up the subject if you're going to throw it away by making odd assumptions? The concept really lacks meaning when you do such a thing.
Finally, please consider using a lower-case t for time, my eye twitches too much otherwise.
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Playing the game a lot and looking to actively improve through experience and getting a great feel for the game is more effective than applying overly complicated economic principles and being more mathematically analytical than is necessary. I feel like I'm in an internet poker forum.
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Didn't anaylze your formulas too much (to be honest it doesn't seem that strange that more workers = more minerals) but:
1. Workers cost supply (roughly 12.5 minerals each)
2. There aren't infinite minerals. It should be fairly easy to get a total mineral count for each map and then average them to determine a limit.
Not sure what effect these things have on your determination, but with a slightly higher probe cost, I'm guessing the extra probes take a little while longer to pay for themselves.
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EPT
